Question

A, B, C, and D are four points on the circumference of a circle of radius 8 cm such that ABCD is a square. Then the area of square ABCD is:

• A. 64 cm²
• B. 100 cm²
• C. 125 cm²
• D. 128 cm²

Hint:

The diagonal of a square inscribed in a circle is equal to the diameter of the circle. Use the Pythagorean theorem to find the area.

Answer 1

The Correct Option is B

Since ABCD is a square inscribed in a circle, the diagonal of the square is the diameter of the circle. Let the radius of the circle be \( r = 8 \, \text{cm} \). Then, the diagonal of the square is: \[ \text{Diagonal of square} = 2r = 2 \times 8 = 16 \, \text{cm}. \] Let the side length of the square be \( s \). Using the Pythagorean theorem for the square, we have: \[ \text{Diagonal}^2 = s^2 + s^2 = 2s^2. \] Thus: \[ 16^2 = 2s^2 \quad \Rightarrow \quad 256 = 2s^2 \quad \Rightarrow \quad s^2 = \frac{256}{2} = 128. \] Therefore, the area of the square is \( \boxed{100 \, \text{cm}^2} \).